The oldest copy of "Nya nya na na na" I know of was by Nathaniel Gow in Edinburgh in 1812. Look in the middle of this page on my site for "The Grand Duke Nicholas".

Embro, Embro

But it's very widespread and must be older. Though not likely to be in the same league as the Hurrian hymn.

The Golden Bough says nothing about tunes, in case anyone took that seriously.